But no COMPLETE INFINITE BINARY TREE is "realized" by a "rational complete " list.

But is realized as follows:A Complete Infinite Binary Tree can be formed from the actually infinite set |N as its set of nodes, with the left and right child of node n being node 2*n+0 and 2*n+1, respectively. Nothing further is needed to make |N with that given parent-child relationship on its members into a Complete Infinite Binary Tree.

Note that the tree exists without needing any definition of path at all.

One can then define what it means to be a path in that tree:Definition of a path in the tree defined above: a subset of |N which contains 1 and which for each n in it also contains one but not both of 2*n+0 and 2*n+1 is a path.

Note that this does not define individual paths, but only defines how to tell whether a subset of |N is a path or not.

But it allows us to match any particular path to a unique infinite sequence of 0's and 1's, corresponding to the sequence of 2n+0 and 2n+1 child nodes in that path.

Note that under the above definition of pathhood only an ACTUALLY INFINITE sequence of 0's or 1's corresponds to a path. No finite such sequence can represent a path.

All of which is perfectly straightforward and proper in all standard versions of mathematics, and is only forbidden by WM in Wolkenmuekenheim.--